How to calculate movement of a radius of one sphere to intersect another sphere and create a specified distance for the intersection
We'll start with the 12 mm radius sphere, note the 12mm chord is in the XY plane, so scaling on the Z axis won't change its size, it will only move the position in the Z direction. Because we can draw a right triangle with hypotenuse 12 (the radius) and leg 6 (half the chord length) we know the remaining side is $6\sqrt{3}$. When we squish the sphere by 90% that side will also squish, so the red dot winds up at $0.9 \cdot 6\sqrt{3}$

Moving onto the smaller, 6.63 mm sphere, we can again draw a right triangle, this time with hypotenuse 6.63 but still with a leg of 6. We need to resort to the distance formula $ k = \sqrt{6.63^2 - 6^2} $ to find the side length because this triangle wasn't special like the last one.

To position the red dots on top of each other, we need to move the green dot so it is in the right place below the red dot: $ \textrm{green dot} = 0.9 \cdot 6\sqrt{3} - \sqrt{6.63^2 - 6^2} $
The final result.